Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(263,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.263"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1,2,-5,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.59066497296.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.2
Root \(-0.115680 - 0.200364i\) of defining polynomial
Character \(\chi\) \(=\) 637.373
Dual form 637.2.g.k.263.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.115680 - 0.200364i) q^{2} -3.32225 q^{3} +(0.973236 - 1.68569i) q^{4} +(1.11568 - 1.93242i) q^{5} +(0.384320 + 0.665661i) q^{6} -0.913059 q^{8} +8.03736 q^{9} -0.516249 q^{10} +3.32225 q^{11} +(-3.23334 + 5.60030i) q^{12} +(3.40300 + 1.19146i) q^{13} +(-3.70657 + 6.41997i) q^{15} +(-1.84085 - 3.18844i) q^{16} +(0.687890 - 1.19146i) q^{17} +(-0.929766 - 1.61040i) q^{18} +3.23531 q^{19} +(-2.17164 - 3.76139i) q^{20} +(-0.384320 - 0.665661i) q^{22} +(-0.419251 - 0.726164i) q^{23} +3.03341 q^{24} +(0.0105144 + 0.0182115i) q^{25} +(-0.154934 - 0.819669i) q^{26} -16.7354 q^{27} +(0.303571 - 0.525800i) q^{29} +1.71511 q^{30} +(-0.857556 - 1.48533i) q^{31} +(-1.33896 + 2.31915i) q^{32} -11.0374 q^{33} -0.318302 q^{34} +(7.82225 - 13.5485i) q^{36} +(-0.776807 - 1.34547i) q^{37} +(-0.374262 - 0.648241i) q^{38} +(-11.3056 - 3.95833i) q^{39} +(-1.01868 + 1.76441i) q^{40} +(4.58892 - 7.94824i) q^{41} +(-0.615680 - 1.06639i) q^{43} +(3.23334 - 5.60030i) q^{44} +(8.96713 - 15.5315i) q^{45} +(-0.0969983 + 0.168006i) q^{46} +(0.814085 - 1.41004i) q^{47} +(6.11577 + 10.5928i) q^{48} +(0.00243263 - 0.00421343i) q^{50} +(-2.28535 + 3.95833i) q^{51} +(5.32036 - 4.57685i) q^{52} +(-4.19803 - 7.27121i) q^{53} +(1.93596 + 3.35318i) q^{54} +(3.70657 - 6.41997i) q^{55} -10.7485 q^{57} -0.140469 q^{58} +(-4.41117 + 7.64037i) q^{59} +(7.21474 + 12.4963i) q^{60} +5.46667 q^{61} +(-0.198405 + 0.343647i) q^{62} -6.74383 q^{64} +(6.09906 - 5.24672i) q^{65} +(1.27681 + 2.21149i) q^{66} -10.1857 q^{67} +(-1.33896 - 2.31915i) q^{68} +(1.39286 + 2.41250i) q^{69} +(2.60714 + 4.51570i) q^{71} -7.33859 q^{72} +(1.98177 + 3.43253i) q^{73} +(-0.179723 + 0.311289i) q^{74} +(-0.0349316 - 0.0605033i) q^{75} +(3.14872 - 5.45375i) q^{76} +(0.514731 + 2.72315i) q^{78} +(-3.22525 + 5.58630i) q^{79} -8.21520 q^{80} +31.4871 q^{81} -2.12339 q^{82} -4.64055 q^{83} +(-1.53493 - 2.65858i) q^{85} +(-0.142444 + 0.246721i) q^{86} +(-1.00854 + 1.74684i) q^{87} -3.03341 q^{88} +(-4.56413 - 7.90530i) q^{89} -4.14929 q^{90} -1.63212 q^{92} +(2.84902 + 4.93464i) q^{93} -0.376695 q^{94} +(3.60957 - 6.25197i) q^{95} +(4.44836 - 7.70479i) q^{96} +(7.67944 + 13.3012i) q^{97} +26.7022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 2 q^{3} - 5 q^{4} + 7 q^{5} + 5 q^{6} - 12 q^{8} + 14 q^{9} - 22 q^{10} - 2 q^{11} - 12 q^{12} + 4 q^{13} - 3 q^{15} - 19 q^{16} + 4 q^{17} + 3 q^{18} + 2 q^{19} + 2 q^{20} - 5 q^{22} + 2 q^{23}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.115680 − 0.200364i −0.0817984 − 0.141679i 0.822224 − 0.569164i \(-0.192733\pi\)
−0.904022 + 0.427485i \(0.859400\pi\)
\(3\) −3.32225 −1.91810 −0.959052 − 0.283231i \(-0.908594\pi\)
−0.959052 + 0.283231i \(0.908594\pi\)
\(4\) 0.973236 − 1.68569i 0.486618 − 0.842847i
\(5\) 1.11568 − 1.93242i 0.498947 − 0.864202i −0.501052 − 0.865417i \(-0.667053\pi\)
0.999999 + 0.00121496i \(0.000386732\pi\)
\(6\) 0.384320 + 0.665661i 0.156898 + 0.271755i
\(7\) 0 0
\(8\) −0.913059 −0.322815
\(9\) 8.03736 2.67912
\(10\) −0.516249 −0.163252
\(11\) 3.32225 1.00170 0.500848 − 0.865535i \(-0.333021\pi\)
0.500848 + 0.865535i \(0.333021\pi\)
\(12\) −3.23334 + 5.60030i −0.933384 + 1.61667i
\(13\) 3.40300 + 1.19146i 0.943823 + 0.330452i
\(14\) 0 0
\(15\) −3.70657 + 6.41997i −0.957033 + 1.65763i
\(16\) −1.84085 − 3.18844i −0.460212 − 0.797111i
\(17\) 0.687890 − 1.19146i 0.166838 − 0.288972i −0.770469 − 0.637478i \(-0.779978\pi\)
0.937306 + 0.348506i \(0.113311\pi\)
\(18\) −0.929766 − 1.61040i −0.219148 − 0.379575i
\(19\) 3.23531 0.742231 0.371116 − 0.928587i \(-0.378975\pi\)
0.371116 + 0.928587i \(0.378975\pi\)
\(20\) −2.17164 − 3.76139i −0.485594 − 0.841073i
\(21\) 0 0
\(22\) −0.384320 − 0.665661i −0.0819372 − 0.141919i
\(23\) −0.419251 − 0.726164i −0.0874199 − 0.151416i 0.819000 − 0.573794i \(-0.194529\pi\)
−0.906420 + 0.422378i \(0.861196\pi\)
\(24\) 3.03341 0.619193
\(25\) 0.0105144 + 0.0182115i 0.00210289 + 0.00364231i
\(26\) −0.154934 − 0.819669i −0.0303851 − 0.160750i
\(27\) −16.7354 −3.22073
\(28\) 0 0
\(29\) 0.303571 − 0.525800i 0.0563717 − 0.0976386i −0.836462 − 0.548024i \(-0.815380\pi\)
0.892834 + 0.450386i \(0.148713\pi\)
\(30\) 1.71511 0.313135
\(31\) −0.857556 − 1.48533i −0.154022 − 0.266773i 0.778681 − 0.627420i \(-0.215889\pi\)
−0.932702 + 0.360647i \(0.882556\pi\)
\(32\) −1.33896 + 2.31915i −0.236697 + 0.409971i
\(33\) −11.0374 −1.92136
\(34\) −0.318302 −0.0545883
\(35\) 0 0
\(36\) 7.82225 − 13.5485i 1.30371 − 2.25809i
\(37\) −0.776807 − 1.34547i −0.127706 − 0.221194i 0.795081 − 0.606503i \(-0.207428\pi\)
−0.922788 + 0.385309i \(0.874095\pi\)
\(38\) −0.374262 − 0.648241i −0.0607133 − 0.105159i
\(39\) −11.3056 − 3.95833i −1.81035 − 0.633841i
\(40\) −1.01868 + 1.76441i −0.161068 + 0.278978i
\(41\) 4.58892 − 7.94824i 0.716668 − 1.24131i −0.245644 − 0.969360i \(-0.578999\pi\)
0.962313 − 0.271946i \(-0.0876672\pi\)
\(42\) 0 0
\(43\) −0.615680 − 1.06639i −0.0938904 − 0.162623i 0.815255 − 0.579103i \(-0.196597\pi\)
−0.909145 + 0.416480i \(0.863264\pi\)
\(44\) 3.23334 − 5.60030i 0.487444 − 0.844277i
\(45\) 8.96713 − 15.5315i 1.33674 − 2.31530i
\(46\) −0.0969983 + 0.168006i −0.0143016 + 0.0247711i
\(47\) 0.814085 − 1.41004i 0.118747 − 0.205675i −0.800525 − 0.599300i \(-0.795446\pi\)
0.919271 + 0.393625i \(0.128779\pi\)
\(48\) 6.11577 + 10.5928i 0.882735 + 1.52894i
\(49\) 0 0
\(50\) 0.00243263 − 0.00421343i 0.000344025 − 0.000595870i
\(51\) −2.28535 + 3.95833i −0.320012 + 0.554278i
\(52\) 5.32036 − 4.57685i 0.737802 − 0.634695i
\(53\) −4.19803 − 7.27121i −0.576644 − 0.998777i −0.995861 − 0.0908909i \(-0.971029\pi\)
0.419217 − 0.907886i \(-0.362305\pi\)
\(54\) 1.93596 + 3.35318i 0.263450 + 0.456310i
\(55\) 3.70657 − 6.41997i 0.499794 − 0.865669i
\(56\) 0 0
\(57\) −10.7485 −1.42368
\(58\) −0.140469 −0.0184445
\(59\) −4.41117 + 7.64037i −0.574285 + 0.994691i 0.421834 + 0.906673i \(0.361387\pi\)
−0.996119 + 0.0880181i \(0.971947\pi\)
\(60\) 7.21474 + 12.4963i 0.931419 + 1.61326i
\(61\) 5.46667 0.699936 0.349968 − 0.936762i \(-0.386193\pi\)
0.349968 + 0.936762i \(0.386193\pi\)
\(62\) −0.198405 + 0.343647i −0.0251974 + 0.0436432i
\(63\) 0 0
\(64\) −6.74383 −0.842979
\(65\) 6.09906 − 5.24672i 0.756495 − 0.650776i
\(66\) 1.27681 + 2.21149i 0.157164 + 0.272216i
\(67\) −10.1857 −1.24439 −0.622193 − 0.782864i \(-0.713758\pi\)
−0.622193 + 0.782864i \(0.713758\pi\)
\(68\) −1.33896 − 2.31915i −0.162373 − 0.281238i
\(69\) 1.39286 + 2.41250i 0.167680 + 0.290431i
\(70\) 0 0
\(71\) 2.60714 + 4.51570i 0.309411 + 0.535915i 0.978234 − 0.207507i \(-0.0665349\pi\)
−0.668823 + 0.743422i \(0.733202\pi\)
\(72\) −7.33859 −0.864861
\(73\) 1.98177 + 3.43253i 0.231949 + 0.401748i 0.958382 − 0.285490i \(-0.0921563\pi\)
−0.726432 + 0.687238i \(0.758823\pi\)
\(74\) −0.179723 + 0.311289i −0.0208923 + 0.0361866i
\(75\) −0.0349316 − 0.0605033i −0.00403355 − 0.00698632i
\(76\) 3.14872 − 5.45375i 0.361183 − 0.625588i
\(77\) 0 0
\(78\) 0.514731 + 2.72315i 0.0582818 + 0.308336i
\(79\) −3.22525 + 5.58630i −0.362869 + 0.628508i −0.988432 − 0.151666i \(-0.951536\pi\)
0.625562 + 0.780174i \(0.284870\pi\)
\(80\) −8.21520 −0.918487
\(81\) 31.4871 3.49857
\(82\) −2.12339 −0.234489
\(83\) −4.64055 −0.509367 −0.254684 − 0.967024i \(-0.581971\pi\)
−0.254684 + 0.967024i \(0.581971\pi\)
\(84\) 0 0
\(85\) −1.53493 − 2.65858i −0.166487 − 0.288363i
\(86\) −0.142444 + 0.246721i −0.0153602 + 0.0266046i
\(87\) −1.00854 + 1.74684i −0.108127 + 0.187281i
\(88\) −3.03341 −0.323363
\(89\) −4.56413 − 7.90530i −0.483797 − 0.837960i 0.516030 − 0.856570i \(-0.327409\pi\)
−0.999827 + 0.0186101i \(0.994076\pi\)
\(90\) −4.14929 −0.437373
\(91\) 0 0
\(92\) −1.63212 −0.170160
\(93\) 2.84902 + 4.93464i 0.295429 + 0.511699i
\(94\) −0.376695 −0.0388531
\(95\) 3.60957 − 6.25197i 0.370334 − 0.641438i
\(96\) 4.44836 − 7.70479i 0.454009 − 0.786367i
\(97\) 7.67944 + 13.3012i 0.779729 + 1.35053i 0.932098 + 0.362206i \(0.117976\pi\)
−0.152369 + 0.988324i \(0.548690\pi\)
\(98\) 0 0
\(99\) 26.7022 2.68367
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.g.k.373.2 8
7.2 even 3 91.2.f.c.22.2 ✓ 8
7.3 odd 6 637.2.h.i.165.3 8
7.4 even 3 637.2.h.h.165.3 8
7.5 odd 6 637.2.f.i.295.2 8
7.6 odd 2 637.2.g.j.373.2 8
13.3 even 3 637.2.h.h.471.3 8
21.2 odd 6 819.2.o.h.568.3 8
28.23 odd 6 1456.2.s.q.113.1 8
91.3 odd 6 637.2.g.j.263.2 8
91.9 even 3 1183.2.a.k.1.3 4
91.16 even 3 91.2.f.c.29.2 yes 8
91.30 even 6 1183.2.a.l.1.2 4
91.55 odd 6 637.2.h.i.471.3 8
91.58 odd 12 1183.2.c.g.337.4 8
91.61 odd 6 8281.2.a.bp.1.3 4
91.68 odd 6 637.2.f.i.393.2 8
91.72 odd 12 1183.2.c.g.337.5 8
91.81 even 3 inner 637.2.g.k.263.2 8
91.82 odd 6 8281.2.a.bt.1.2 4
273.107 odd 6 819.2.o.h.757.3 8
364.107 odd 6 1456.2.s.q.1121.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.2 ✓ 8 7.2 even 3
91.2.f.c.29.2 yes 8 91.16 even 3
637.2.f.i.295.2 8 7.5 odd 6
637.2.f.i.393.2 8 91.68 odd 6
637.2.g.j.263.2 8 91.3 odd 6
637.2.g.j.373.2 8 7.6 odd 2
637.2.g.k.263.2 8 91.81 even 3 inner
637.2.g.k.373.2 8 1.1 even 1 trivial
637.2.h.h.165.3 8 7.4 even 3
637.2.h.h.471.3 8 13.3 even 3
637.2.h.i.165.3 8 7.3 odd 6
637.2.h.i.471.3 8 91.55 odd 6
819.2.o.h.568.3 8 21.2 odd 6
819.2.o.h.757.3 8 273.107 odd 6
1183.2.a.k.1.3 4 91.9 even 3
1183.2.a.l.1.2 4 91.30 even 6
1183.2.c.g.337.4 8 91.58 odd 12
1183.2.c.g.337.5 8 91.72 odd 12
1456.2.s.q.113.1 8 28.23 odd 6
1456.2.s.q.1121.1 8 364.107 odd 6
8281.2.a.bp.1.3 4 91.61 odd 6
8281.2.a.bt.1.2 4 91.82 odd 6